An Introduction to Lie Groups and Symplectic Geometry by Bryant R.L.

By Bryant R.L.

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What can you say about the quotient g/[g, g]? 22. Show that, for a connected Lie group G, a connected Lie subgroup H is normal if and only if h is an ideal of g. ) 23. For any Lie algebra g, let z(g) ⊂ g denote the kernel of the homomorphism ad: g → gl(g). Use Theorem 2 and Exercise 16 to prove Theorem 4 for any Lie algebra g for which z(g) = 0. ) Show also that if g is the Lie algebra of the connected Lie group G, then the connected Lie subgroup Z(g) ⊂ G which corresponds to z(g) lies in the center of G.

4 36 (i) Show that κ is symmetric and, if g is the Lie algebra of a Lie group G, then κ is Ad-invariant: κ Ad(g)x, Ad(g)y = κ(x, y) = κ(y, x). Show also that κ [z, x], y = −κ x, [z, y] . A Lie algebra g is said to be semi-simple if κ is a non-degenerate bilinear form on g. (ii) Show that, of all the 2- and 3-dimensional Lie algebras, only so(3) and sl(2, R) are semi-simple. (iii) Show that if h ⊂ g is an ideal in a semi-simple Lie algebra g, then the Killing form of h as an algebra is equal to the restriction of the Killing form of g to h.

Show that its Lie algebra is der(g) = {a ∈ End(g) | a [x, y] = a(x), y + x, a(y) (ii) (iii) (iv) (v) for all x, y ∈ g}. ) Show that if G is a connected and simply connected Lie group with Lie algebra g, then the group of (Lie) automorphisms of G is isomorphic to Aut(g). Show that ad: g → End(g) actually has its image in der(g), and that this image is an ideal in der(g). What is the interpretation of this fact in terms of “inner” and “outer” automorphisms of G? ) Show that if the Killing form of g is non-degenerate, then [g, g] = g.

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